You don't mention what kind of line it is, but the most common are plastic or copper. I would replace the leaky line with one that has a braided steel covering. These are similar to those used for connecting toilets and sinks. They are quite durable and will flex without kinking. Copper is good in many ways, but do tend to kink and develop leaks around the connection to the valve and/or fridge when you pull the fridge out for cleaning.

Gary is dead on right, and if you go buy one at the local hardware store it is usally about six foot long and has 1/4" compression nuts on each end and will actually say ice maker supply on the package.